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# Krasnoselskii-type algorithm for fixed points of multi-valued strictly pseudo-contractive mappings

- Charles E Chidume
^{1}Email author, - Chu-Chu O Chidume
^{2}, - Ngalla Djitte
^{1, 3}and - Maaruf S Minjibir
^{1, 4}

**2013**:58

https://doi.org/10.1186/1687-1812-2013-58

© Chidume et al.; licensee Springer 2013

**Received:**8 August 2012**Accepted:**10 February 2013**Published:**14 March 2013

## Abstract

Let $q>1$ and let *K* be a nonempty, closed and convex subset of a *q*-uniformly smooth real Banach space *E*. Let $T:K\to CB(K)$ be a multi-valued strictly pseudo-contractive map with a nonempty fixed point set. A Krasnoselskii-type iteration sequence $\{{x}_{n}\}$ is constructed and proved to be an approximate fixed point sequence of *T*, *i.e.*, ${lim}_{n\to \mathrm{\infty}}d({x}_{n},T{x}_{n})=0$. This result is then applied to prove strong convergence theorems for a fixed point of *T* under additional appropriate conditions. Our theorems improve several important well-known results.

**MSC:**47H04, 47H06, 47H15, 47H17, 47J25.

## Keywords

*k*-strictly pseudo-contractive mappings- multi-valued mappings
*q*-uniformly smooth spaces

## 1 Introduction

For decades, the study of fixed point theory for *multi-valued nonlinear mappings* has attracted, and continues to attract, the interest of several well-known mathematicians (see, for example, Brouwer [1], Kakutani [2], Nash [3, 4], Geanakoplos [5], Nadla [6], Downing and Kirk [7]).

Interest in the study of fixed point theory for multi-valued maps stems, perhaps, mainly from the fact that many problems in some areas of mathematics such as in *Game Theory and Market Economy* and in *Non-Smooth Differential Equations* can be written as fixed point problems for multi-valued maps. We describe briefly the connection of fixed point theory for multi-valued mappings and these applications.

### Game theory and market economy

In game theory and market economy, the existence of equilibrium was uniformly obtained by the application of a fixed point theorem. In fact, Nash [3, 4] showed the existence of equilibria for non-cooperative static games as a direct consequence of Brouwer [1] or Kakutani [2] fixed point theorem. More precisely, under some regularity conditions, given a game, there always exists a *multi-valued map* whose fixed points coincide with the equilibrium points of the game. A model example of such an application is the *Nash equilibrium theorem* (see, *e.g.*, [3]).

*N*players denoted by

*n*, $n=1,\dots ,N$, where ${K}_{n}\subset {\mathbb{R}}^{{m}_{n}}$ is the set of possible strategies of the

*n*th player and is assumed to be nonempty, compact and convex and ${u}_{n}:K:={K}_{1}\times {K}_{2}\cdots \times {K}_{N}\to \mathbb{R}$ is the payoff (or gain function) of the player

*n*and is assumed to be continuous. The player

*n*can take

*individual actions*, represented by a vector ${\sigma}_{n}\in {K}_{n}$. All players together can take a

*collective action*, which is a combined vector $\sigma =({\sigma}_{1},{\sigma}_{2},\dots ,{\sigma}_{N})$. For each

*n*, $\sigma \in K$ and ${z}_{n}\in {K}_{n}$, we will use the following standard notations:

*n*th player to maximize his gain

*under the condition*that the

*remaining players*have chosen their strategies ${\sigma}_{-n}$ if and only if

**Definition**A collective action $\overline{\sigma}=({\overline{\sigma}}_{1},\dots ,{\overline{\sigma}}_{N})\in K$ is called a

*Nash equilibrium point*if, for each

*n*, ${\overline{\sigma}}_{n}$ is the best response for the

*n*th player to the action ${\overline{\sigma}}_{-n}$ made by the remaining players. That is, for each

*n*,

From the point of view of social recognition, game theory is perhaps the most successful area of application of fixed point theory of multi-valued mappings. However, it has been remarked that the applications of this theory to equilibrium are mostly static: they enhance understanding of conditions under which equilibrium may be achieved but do not indicate how to construct a process starting from a non-equilibrium point and convergent to an equilibrium solution. This is part of the problem that is being addressed by iterative methods for a fixed point of multi-valued mappings.

### Non-smooth differential equations

The mainstream of applications of fixed point theory for multi-valued maps has been initially motivated by the problem of differential equations (DEs) with discontinuous right-hand sides which gave birth to the existence theory of differential inclusion (DI). Here is a simple model for this type of application.

*t*, one looks for

*solutions*in the sense of Filippov [8] which are solutions of the differential inclusion

*multi-valued Nemystkii operator*defined by

One can see that problem (1.4) reduces to the fixed point problem: $u\in Tu$.

*E*be a real normed linear space of dimension ≥2. The

*modulus of smoothness*of

*E*, ${\rho}_{E}$, is defined by

*E*is called

*uniformly smooth*if

*e.g.*, [13], p.16, [14]) that ${\rho}_{E}$ is nondecreasing. If there exist a constant $c>0$ and a real number $q>1$ such that ${\rho}_{E}(\tau )\le c{\tau}^{q}$, then

*E*is said to be

*q*-

*uniformly smooth*. Typical examples of such spaces are the ${L}_{p}$, ${\ell}_{p}$ and ${W}_{p}^{m}$ spaces for $1<p<\mathrm{\infty}$, where

*generalized duality mapping*from

*E*to ${2}^{{E}^{\ast}}$ defined by

where $\u3008\cdot ,\cdot \u3009$ denotes the generalized duality pairing. ${J}_{2}$ is called the *normalized duality mapping* and is denoted by *J*. It is well known that if *E* is smooth, ${J}_{q}$ is single-valued.

Every uniformly smooth space has a uniformly Gâteaux differentiable norm (see, *e.g.*, [13], p.17).

*K*be a nonempty subset of

*E*. The set

*K*is called

*proximinal*(see,

*e.g.*, [15–17]) if for each $x\in E$, there exists $u\in K$ such that

*K*, respectively. The

*Hausdorff metric*on $CB(K)$ is defined by

for all $A,B\in CB(K)$. Let $T:D(T)\subseteq E\to CB(E)$ be a *multi-valued mapping* on *E*. A point $x\in D(T)$ is called a *fixed point of* *T* if $x\in Tx$. The fixed point set of *T* is denoted by $F(T):=\{x\in D(T):x\in Tx\}$.

*L*-

*Lipschitzian*if there exists $L>0$ such that

When $L\in (0,1)$ in (1.6), we say that *T* is a *contraction*, and *T* is called *nonexpansive* if $L=1$.

**Definition 1.1**Let

*K*be a nonempty subset of a real Hilbert space

*H*. A map $T:K\to H$ is called

*k*-

*strictly pseudo-contractive*if there exists $k\in (0,1)$ such that

Browder and Petryshyn [18] introduced and studied the class of strictly pseudo-contractive maps as an important generalization of the class of nonexpansive maps (mappings $T:K\to K$ satisfying $\parallel Tx-Ty\parallel \le \parallel x-y\parallel \phantom{\rule{0.25em}{0ex}}\mathrm{\forall}x,y\in K$). It is trivial to see that every nonexpansive map is strictly pseudo-contractive.

Motivated by this, Chidume *et al.*[19] introduced the class of *multi-valued strictly pseudo-contractive* maps defined on a real Hilbert space *H* as follows.

**Definition 1.2**A multi-valued map $T:D(T)\subset H\to CB(H)$ is called

*k*-

*strictly pseudo-contractive*if there exists $k\in (0,1)$ such that for all $x,y\in D(T)$,

They then proved convergence theorems for approximating fixed points of multi-valued strictly pseudo-contractive maps (see [19]) which extend recent results from the class of multi-valued *nonexpansive maps* to the more general and important class of multi-valued *strictly pseudo-contractive maps*.

Single-valued strictly pseudo-contractive maps have also been defined and studied *in real Banach spaces, which are much more general than Hilbert spaces*.

**Definition 1.3**Let

*K*be a nonempty subset of a real normed space

*E*. A map $T:K\to E$ is called

*k*-

*strictly pseudo-contractive*(see,

*e.g.*, [13], p.145, [18]) if there exists $k\in (0,1)$ such that for all $x,y\in K$, there exists $j(x-y)\in J(x-y)$ such that

In this paper, we define multi-valued strictly pseudo-contractive maps in *arbitrary normed space* *E* as follows.

**Definition 1.4**A multi-valued map $T:D(T)\subset E\to CB(E)$ is called

*k*-

*strictly pseudo-contractive*if there exists $k\in (0,1)$ such that for all $x,y\in D(T)$,

where $A:=I-T$ and *I* is the identity map on *E*.

We observe that if *T* is single-valued, then inequality (1.10) reduces to (1.9).

Several papers deal with the problem of approximating fixed points of *multi-valued nonexpansive mappings defined on Hilbert spaces* (see, for example, Sastry and Babu [15], Panyanak [16], Song and Wong [17], Khan *et al.*[20], Abbas *et al.*[21] and the references contained therein) and their generalizations (see, *e.g.*, Chidume *et al.*[19] and the references contained therein).

Chidume *et al.*[19] proved the following theorem for multi-valued *k*-strictly pseudo-contractive mappings defined on *real Hilbert spaces*.

**Theorem CCDM** (Theorem 3.2 [19])

*Let*

*K*

*be a nonempty*,

*closed and convex subset of a real Hilbert space*

*H*.

*Suppose that*$T:K\to CB(K)$

*is a multi*-

*valued*

*k*-

*strictly pseudo*-

*contractive mapping such that*$F(T)\ne \mathrm{\varnothing}$.

*Assume that*$Tp=\{p\}$

*for all*$p\in F(T)$.

*Let*$\{{x}_{n}\}$

*be a sequence defined iteratively from*${x}_{0}\in K$

*by*

*where*${y}_{n}\in T{x}_{n}$*and*$\lambda \in (0,1-k)$. *Then*${lim}_{n\to \mathrm{\infty}}d({x}_{n},T{x}_{n})=0$.

Using Theorem CCDM, Chidume *et al.* proved several convergence theorems for the approximation of fixed points of strictly pseudo-contractive maps under various additional mild compactness-type conditions either on the operator *T* or on the domain of *T*. The theorems proved in [19] are significant generalizations of several important results on Hilbert spaces (see, *e.g.*, [19]).

Our purpose in this paper is to extend Theorem CCDM and other related results in [19], using Definition 1.4, from Hilbert spaces to the much more general class of *q*-*uniformly smooth real Banach spaces*. As we have noted, theses spaces include the ${L}_{p}$, ${l}_{p}$ and ${W}^{m,p}$ spaces, $1<p<\mathrm{\infty}$ and $m\ge 1$. Finally, we give important examples of multi-valued maps satisfying the conditions of our theorems.

## 2 Preliminaries

In the sequel, we need the following definitions and results.

**Definition 2.1** Let *E* be a real Banach space and *T* be a multi-valued mapping. The multi-valued map $(I-T)$ is said to be *strongly demiclosed* at 0 (see, *e.g.*, [22]) if for any sequence $\{{x}_{n}\}\subseteq D(T)$ such that $\{{x}_{n}\}$ converges strongly to ${x}^{\ast}$ and $d({x}_{n},T{x}_{n})$ converges to 0, then $d({x}^{\ast},T{x}^{\ast})=0$.

**Lemma 2.2**[19]

*Let*

*E*

*be a reflexive real Banach space and let*$A,B\in CB(X)$.

*Assume that*

*B*

*is weakly closed*.

*Then*,

*for every*$a\in A$,

*there exists*$b\in B$

*such that*

**Proposition 2.3** *Let* *K* *be a nonempty subset of a real Banach space* *E* *and let*$T:K\to CB(K)$*be a multi*-*valued* *k*-*strictly pseudo*-*contractive mapping*. *Assume that for every*$x\in K$, *Tx* *is weakly closed*. *Then* *T* *is Lipschitzian*.

*Proof*We first observe that for any $x\in D(T)$, the set

*Tx*is weakly closed if and only if the set

*Ax*is weakly closed. Now, let $x,y\in D(T)$ and $u\in Ax$. From Lemma 2.2, there exists $v\in Ay$ such that

*T*is

*k*-strictly pseudo-contractive and inequality (2.2), we have

Therefore, *T* is ${L}_{k}$-Lipschitzian. □

**Remark 1** We note that *for a single-valued map* *T*, for each $x\in D(T)$, the set *Tx* is always weakly closed.

**Lemma 2.4**

*Let*$q>1$,

*E*

*be a*

*q*-

*uniformly smooth real Banach space*, $k\in (0,1)$.

*Suppose*$T:D(T)\subset E\to CB(E)$

*is a multi*-

*valued map with*$F(T)\ne \mathrm{\varnothing}$,

*and for all*$x\in D(T)$, ${x}^{\ast}\in F(T)$,

*where*$A:=I-T$,

*I*

*is the identity map on*

*E*.

*If*$T{x}^{\ast}=\{{x}^{\ast}\}$

*for all*${x}^{\ast}\in F(T)$,

*then the following inequality holds*:

*Proof*Let $x\in D(T)$, $u\in Ax$, ${x}^{\ast}\in F(T)$. Then, from inequality (2.5), the definition of the Hausdorff metric and the assumption that $T{x}^{\ast}=\{{x}^{\ast}\}$, we have

$D(Ax,A{x}^{\ast})\ge \parallel x-{x}^{\ast}\parallel $ since $T{x}^{\ast}=\{{x}^{\ast}\}$. This completes the proof. □

**Lemma 2.5** *Let* *K* *be a nonempty closed subset of a real Banach space* *E* *and let*$T:K\to P(K)$*be a* *k*-*strictly pseudo*-*contractive mapping*. *Assume that for every*$x\in K$, *Tx* *is weakly closed*. *Then*$(I-T)$*is strongly demiclosed at zero*.

*Proof*Let $\{{x}_{n}\}\subseteq K$ be such that ${x}_{n}\to x$ and $d({x}_{n},T{x}_{n})\to 0$ as $n\to \mathrm{\infty}$. Since

*K*is closed, we have that $x\in K$. Since, for every

*n*, $T{x}_{n}$ is proximinal, let ${y}_{n}\in T{x}_{n}$ such that $\parallel {x}_{n}-{y}_{n}\parallel =d({x}_{n},T{x}_{n})$. Using Lemma 2.2, for each

*n*, there exists ${z}_{n}\in Tx$ such that

Taking limit as $n\to \mathrm{\infty}$, we have that $d(x,Tx)=0$. Therefore $x\in Tx$. The proof is completed. □

**Lemma 2.6**[23]

*Let*$q>1$

*and*

*E*

*be a smooth real Banach space*.

*Then the following are equivalent*:

- (i)
*E**is**q*-*uniformly smooth*. - (ii)
*There exists a constant*${d}_{q}>0$*such that for all*$x,y\in E$,${\parallel x+y\parallel}^{q}\le {\parallel x\parallel}^{q}+q\u3008y,{j}_{q}(x)\u3009+{d}_{q}{\parallel y\parallel}^{q}.$ - (iii)
*There exists a constant*${c}_{q}>0$*such that for all*$x,y\in E$*and*$\lambda \in [0,1]$,${\parallel (1-\lambda )x+\lambda y\parallel}^{q}\ge (1-\lambda ){\parallel x\parallel}^{q}+\lambda {\parallel y\parallel}^{q}-{w}_{q}(\lambda ){c}_{q}{\parallel x-y\parallel}^{q},$

*where*${w}_{q}(\lambda ):={\lambda}^{q}(1-\lambda )+\lambda {(1-\lambda )}^{q}$.

From now on, ${d}_{q}$ denotes the constant that appeared in Lemma 2.6. Let $\mu :=min\{1,{(\frac{q{k}^{q-1}}{{d}_{q}})}^{\frac{1}{q-1}}\}$.

## 3 Main results

We prove the following theorem.

**Theorem 3.1**

*Let*$q>1$

*be a real number and*

*K*

*be a nonempty*,

*closed and convex subset of a*

*q*-

*uniformly smooth real Banach space*

*E*.

*Suppose that*$T:K\to CB(K)$

*is a multi*-

*valued*

*k*-

*strictly pseudo*-

*contractive mapping such that*$F(T)\ne \mathrm{\varnothing}$

*and such that*$Tp=\{p\}$

*for all*$p\in F(T)$.

*For arbitrary*${x}_{1}\in K$

*and*$\lambda \in (0,\mu )$,

*let*$\{{x}_{n}\}$

*be a sequence defined iteratively by*

*where*${y}_{n}\in T{x}_{n}$. *Then*${lim}_{n\to \mathrm{\infty}}d({x}_{n},T{x}_{n})=0$.

*Proof*Let ${x}^{\ast}\in F(T)$. Then, using the recursion formula (3.1), Lemmas 2.6 and 2.4, we have

Hence, ${lim}_{n\to \mathrm{\infty}}\parallel {x}_{n}-{y}_{n}\parallel =0$. Since ${y}_{n}\in T{x}_{n}$, we have that ${lim}_{n\to \mathrm{\infty}}d({x}_{n},T{x}_{n})=0$. □

A mapping $T:K\to CB(K)$ is called *hemicompact* if, for any sequence $\{{x}_{n}\}$ in *K* such that $d({x}_{n},T{x}_{n})\to 0$ as $n\to \mathrm{\infty}$, there exists a subsequence $\{{x}_{{n}_{j}}\}$ of $\{{x}_{n}\}$ such that ${x}_{{n}_{j}}\to p\in K$. We note that if *K* is compact, then every multi-valued mapping $T:K\to CB(K)$ is hemicompact.

We now prove the following corollaries of Theorem 3.1.

**Corollary 3.2**

*Let*$q>1$

*be a real number and*

*K*

*be a nonempty*,

*closed and convex subset of a*

*q*-

*uniformly smooth real Banach space*

*E*.

*Let*$T:K\to CB(K)$

*be a multi*-

*valued*

*k*-

*strictly pseudo*-

*contractive mapping with*$F(T)\ne \mathrm{\varnothing}$

*and such that*$Tp=\{p\}$

*for all*$p\in F(T)$.

*Suppose that*

*T*

*is continuous and hemicompact*.

*Let*$\{{x}_{n}\}$

*be a sequence defined iteratively from*${x}_{1}\in K$

*by*

*where*${y}_{n}\in T{x}_{n}$*and*$\lambda \in (0,\mu )$. *Then the sequence*$\{{x}_{n}\}$*converges strongly to a fixed point of T*.

*Proof* From Theorem 3.1, we have ${lim}_{n\to \mathrm{\infty}}d({x}_{n},T{x}_{n})=0$. Since *T* is hemicompact, there exists a subsequence $\{{x}_{{n}_{j}}\}$ of $\{{x}_{n}\}$ such that ${x}_{{n}_{j}}\to p$ for some $p\in K$. Since *T* is continuous, we have $d({x}_{{n}_{j}},T{x}_{{n}_{j}})\to d(p,Tp)$. Therefore, $d(p,Tp)=0$ and so $p\in F(T)$. Setting ${x}^{\ast}=p$ in the proof of Theorem 3.1, it follows from inequality (3.2) that ${lim}_{n\to \mathrm{\infty}}\parallel {x}_{n}-p\parallel $ exists. So, $\{{x}_{n}\}$ converges strongly to *p*. This completes the proof. □

**Corollary 3.3**

*Let*$q>1$

*be a real number and*

*K*

*be a nonempty*,

*compact and convex subset of a*

*q*-

*uniformly smooth real Banach space*

*E*.

*Let*$T:K\to CB(K)$

*be a multi*-

*valued*

*k*-

*strictly pseudo*-

*contractive mapping with*$F(T)\ne \mathrm{\varnothing}$

*and such that*$Tp=\{p\}$

*for all*$p\in F(T)$.

*Suppose that*

*T*

*is continuous*.

*Let*$\{{x}_{n}\}$

*be a sequence defined iteratively from*${x}_{1}\in K$

*by*

*where*${y}_{n}\in T{x}_{n}$*and*$\lambda \in (0,\mu )$. *Then the sequence*$\{{x}_{n}\}$*converges strongly to a fixed point of T*.

*Proof* Observing that if *K* is compact, every map $T:K\to CB(K)$ is hemicompact, the proof follows from Corollary 3.2. □

**Remark 2** In Corollary 3.2, the continuity assumption on *T* can be dispensed if we assume that for every $x\in K$, the set *Tx* is proximinal and weakly closed. In fact, we have the following result.

**Corollary 3.4**

*Let*$q>1$

*be a real number and*

*K*

*be a nonempty*,

*closed and convex subset of a*

*q*-

*uniformly smooth real Banach space*

*E*.

*Let*$T:K\to CB(K)$

*be a multi*-

*valued*

*k*-

*strictly pseudo*-

*contractive mapping with*$F(T)\ne \mathrm{\varnothing}$

*and such that for every*$x\in K$,

*Tx*

*is weakly closed and*$Tp=\{p\}$

*for all*$p\in F(T)$.

*Suppose that*

*T*

*is hemicompact*.

*Let*$\{{x}_{n}\}$

*be a sequence defined iteratively from*${x}_{1}\in K$

*by*

*where*${y}_{n}\in T{x}_{n}$*and*$\lambda \in (0,\mu )$. *Then the sequence*$\{{x}_{n}\}$*converges strongly to a fixed point of T*.

*Proof* Following the same arguments as in the proof of Corollary 3.2, we have ${x}_{{n}_{j}}\to p$ and ${lim}_{n\to \mathrm{\infty}}d({x}_{{n}_{j}},T{x}_{{n}_{j}})=0$. Furthermore, from Lemma 2.5, $(I-T)$ is strongly demiclosed at zero. It then follows that $p\in Tp$. Setting ${x}^{\ast}=p$ and following the same computations as in the proof of Theorem 3.1, we have from inequality (3.2) that $lim\parallel {x}_{n}-p\parallel $ exists. Since $\{{x}_{{n}_{j}}\}$ converges strongly to *p*, it follows that $\{{x}_{n}\}$ converges strongly to $p\in F(T)$. The proof is completed. □

Convergence theorems have been proved in real Hilbert spaces for multi-valued nonexpansive mappings *T* under the assumption that *T* satisfies Condition (I) (see, *e.g.*, [16, 24]). The following corollary extends such theorems to multi-valued strictly pseudo-contractive maps and to *q*-uniformly smooth real Banach spaces.

**Corollary 3.5**

*Let*$q>1$

*be a real number and*

*K*

*be a nonempty*,

*closed and convex subset of a*

*q*-

*uniformly smooth real Banach space*

*E*.

*Let*$T:K\to P(K)$

*be a multi*-

*valued*

*k*-

*strictly pseudo*-

*contractive mapping with*$F(T)\ne \mathrm{\varnothing}$

*and such that for every*$x\in K$,

*Tx*

*is weakly closed and*$Tp=\{p\}$

*for all*$p\in F(T)$.

*Suppose that*

*T*

*satisfies Condition*(I).

*Let*$\{{x}_{n}\}$

*be a sequence defined iteratively from*${x}_{1}\in K$

*by*

*where*${y}_{n}\in T{x}_{n}$*and*$\lambda \in (0,\mu )$. *Then the sequence*$\{{x}_{n}\}$*converges strongly to a fixed point of T*.

*Proof*From Theorem 3.1, we have ${lim}_{n\to \mathrm{\infty}}d({x}_{n},T{x}_{n})=0$. Using the fact that

*T*satisfies Condition (I), it follows that ${lim}_{n\to \mathrm{\infty}}f(d({x}_{n},F(T)))=0$. Thus there exist a subsequence $\{{x}_{{n}_{j}}\}$ of $\{{x}_{n}\}$ and a sequence $\{{p}_{j}\}\subset F(T)$ such that

*K*. Notice that

*K*and thus converges strongly to some $p\in K$. Using the fact that

*T*is

*L*-Lipschitzian and ${p}_{j}\to p$, we have

so that $d(p,Tp)=0$ and thus $p\in Tp$. Therefore, $p\in F(T)$ and $\{{x}_{{n}_{j}}\}$ converges strongly to *p*. Setting ${x}^{\ast}=p$ in the proof of Theorem 3.1, it follows from inequality (3.2) that ${lim}_{n\to \mathrm{\infty}}\parallel {x}_{n}-p\parallel $ exists. So, $\{{x}_{n}\}$ converges strongly to *p*. This completes the proof. □

**Corollary 3.6**

*Let*$q>1$

*be a real number and*

*K*

*be a nonempty*,

*compact and convex subset of a*

*q*

*uniformly smooth real Banach space*

*E*.

*Let*$T:K\to P(K)$

*be a multi*-

*valued*

*k*-

*strictly pseudo*-

*contractive mapping with*$F(T)\ne \mathrm{\varnothing}$

*and such that for every*$x\in K$,

*the set*

*Tx*

*is weakly closed and*$Tp=\{p\}$

*for all*$p\in F(T)$.

*Let*$\{{x}_{n}\}$

*be a sequence defined iteratively from*${x}_{1}\in K$

*by*

*where*${y}_{n}\in T{x}_{n}$*and*$\lambda \in (0,\mu )$. *Then the sequence*$\{{x}_{n}\}$*converges strongly to a fixed point of T*.

*Proof* From Theorem 3.1, we have ${lim}_{n\to \mathrm{\infty}}d({x}_{n},T{x}_{n})=0$. Since $\{{x}_{n}\}\subseteq K$ and *K* is compact, $\{{x}_{n}\}$ has a subsequence $\{{x}_{{n}_{j}}\}$ that converges strongly to some $p\in K$. Furthermore, from Lemma 2.5, $(I-T)$ is strongly demiclosed at zero. It then follows that $p\in Tp$. Setting ${x}^{\ast}=p$ and following the same arguments as in the proof of Theorem 3.1, we have from inequality (3.2) that $lim\parallel {x}_{n}-p\parallel $ exists. Since $\{{x}_{{n}_{j}}\}$ converges strongly to *q*, it follows that $\{{x}_{n}\}$ converges strongly to $p\in F(T)$. This completes the proof. □

**Corollary 3.7**

*Let*$q>1$

*be a real number and*

*K*

*be a nonempty compact convex subset of a*

*q*

*uniformly smooth real Banach space*

*E*.

*Let*$T:K\to P(K)$

*be a multi*-

*valued nonexpansive mapping*.

*Assume that*$Tp=\{p\}$

*for all*$p\in F(T)$.

*Let*$\{{x}_{n}\}$

*be a sequence defined iteratively from*${x}_{1}\in K$,

*where*${y}_{n}\in T{x}_{n}$*and*$\lambda \in (0,\mu )$. *Then the sequence*$\{{x}_{n}\}$*converges strongly to a fixed point of T*.

**Remark 3** The recursion formula (3.1) of Theorem 3.1 is of the Krasnoselkii type (see, *e.g.*, [25]) and is known to be superior to the recursion formula of the Mann algorithm (see, *e.g.*, Mann [26]) in the following sense: (i) The recursion formula (3.1) requires less computation time than the formula of the Mann algorithm because the parameter *λ* in formula (3.1) is fixed in $(0,1-k)$, whereas in the algorithm of Mann, *λ* is replaced by a sequence $\{{c}_{n}\}$ in $(0,1)$ satisfying the following conditions: ${\sum}_{n=1}^{\mathrm{\infty}}{c}_{n}=\mathrm{\infty}$, $lim{c}_{n}=0$. The ${c}_{n}$ must be computed at each step of the iteration process. (ii) The Krasnoselskii-type algorithm usually yields rate of convergence as fast as that of a geometric progression, whereas the Mann algorithm usually has order of convergence of the form $o(1/n)$.

**Remark 4**In [24], the authors replace the condition $Tp=\{p\}$$\mathrm{\forall}p\in F(T)$ with the following restriction on the sequence ${y}_{n}:{y}_{n}\in {P}_{T}{x}_{n}$,

*i.e.*, ${y}_{n}\in T{x}_{n}$ and $\parallel {y}_{n}-{x}_{n}\parallel =d({x}_{n},T{x}_{n})$. We observe that if, for example, the set $T{x}_{n}$ is a closed and convex subset of a real Hilbert space, then ${y}_{n}$ is

*unique*and is characterized by

Since this ${y}_{n}$ has to be computed at each step of the iteration process, this makes the recursion formula difficult to use in any possible application.

**Remark 5** The addition of *bounded* error terms to the recursion formula (3.1) leads to no generalization.

**Remark 6** Our theorems in this paper are important generalizations of several important recent results in the following sense: (i) Our theorems extend results proved for multi-valued *nonexpansive* mappings in *real Hilbert spaces* (see, *e.g.*, [15–17, 20, 21]) to a much larger class of multi-valued *strictly pseudo-contractive mappings* and in a much larger class of *q*-*uniformly smooth real Banach spaces*. (ii) Our theorems are proved with the superior Krasnoselskii-type algorithm.

We give examples of multi-valued maps where, for each $x\in K$, the set *Tx* is proximinal and weakly closed.

**Example 1**Let $f:\mathbb{R}\to \mathbb{R}$ be an increasing function. Define $T:\mathbb{R}\to {2}^{\mathbb{R}}$ by

where $f(x-):={lim}_{y\to {x}^{-}}f(y)$ and $f(x+):={lim}_{y\to {x}^{+}}f(y)$. For every $x\in \mathbb{R}$, *Tx* is either a singleton or a closed and bounded interval. Therefore, *Tx* is always weakly closed and convex. Hence, for every $x\in \mathbb{R}$, the set *Tx* is proximinal and weakly closed.

**Example 2**Let

*H*be a real Hilbert space and $f:H\to \mathbb{R}$ be a convex continuous function. Let $T:H\to {2}^{H}$ be the multi-valued map defined by

*subdifferential*of

*f*at

*x*and is defined by

It is well known that for every $x\in H$, $\partial f(x)$ is nonempty, weakly closed and convex. Therefore, since *H* is a real Hilbert space, it then follows that for every $x\in H$, the set *Tx* is proximinal and weakly closed. The subdifferential has deep connection with convex optimization problems.

The condition $Tp=\{p\}$ for all $p\in F(T)$, which is imposed in all our theorems of this paper, can actually be replaced by another condition (see, e.g., Shahzad and Zegeye [24]). This is done in Theorem 3.9.

*K*be a nonempty, closed and convex subset of a real Hilbert space, $T:K\to P(K)$ be a multi-valued map and ${P}_{T}:K\to CB(K)$ be defined by

We will need the following result.

**Lemma 3.8** (Song and Cho [27])

*Let*

*K*

*be a nonempty subset of a real Banach space and*$T:K\to P(K)$

*be a multi*-

*valued map*.

*Then the following are equivalent*:

- (i)
${x}^{\ast}\in F(T)$;

- (ii)
${P}_{T}({x}^{\ast})=\{{x}^{\ast}\}$;

- (iii)
${x}^{\ast}\in F({P}_{T})$.

*Moreover*, $F(T)=F({P}_{T})$.

**Remark 7** We observe from Lemma 3.8 that if $T:K\to P(K)$ is *any multi-valued map* with $F(T)\ne \mathrm{\varnothing}$, then the corresponding multi-valued map ${P}_{T}$ satisfies ${P}_{T}(p)=\{p\}$ for all $p\in F({P}_{T})$, the condition imposed in all our theorems and corollaries. Consequently, the examples of multi-valued maps $T:K\to CB(K)$ satisfying the condition $Tp=\{p\}$ for all $p\in F(T)$ abound.

**Theorem 3.9**

*Let*$q>1$

*be a real number and*

*K*

*be a nonempty*,

*closed and convex subset of a*

*q*-

*uniformly smooth real Banach space*

*E*.

*Suppose that*$T:K\to CB(K)$

*is a multi*-

*valued mapping such that*$F(T)\ne \mathrm{\varnothing}$.

*Assume that*${P}_{T}$

*is*

*k*-

*strictly pseudo*-

*contractive*.

*For arbitrary*${x}_{1}\in K$

*and*$\lambda \in (0,\mu )$,

*let*$\{{x}_{n}\}$

*be a sequence defined iteratively by*

*where*${y}_{n}\in {P}_{T}({x}_{n})$. *Then*${lim}_{n\to \mathrm{\infty}}d({x}_{n},T{x}_{n})=0$.

We conclude this paper with an example of a multi-valued map *T* for which ${P}_{T}$ is *k*-strictly pseudo-contractive, the condition assumed in Theorem 3.9. Trivially, every nonexpansive map is strictly pseudo-contractive.

**Example 3**Let $E=\mathbb{R}$ with the usual metric and $T:\mathbb{R}\to CB(\mathbb{R})$ be the multi-valued map defined by

Then ${P}_{T}$ is strictly pseudo-contractive. In fact, ${P}_{T}(x)=\{\frac{x}{2}\}$ for all $x\in \mathbb{R}$.

## Declarations

### Acknowledgements

The authors thank the referees for their comments and remarks that helped to improve the presentation of this paper.

## Authors’ Affiliations

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